Abstract
LetK be a field of characteristicp>0 andF/K be an algebraic function field. We obtain several results on Galois extensionsE/F with an elementary Abelian Galois group of orderpn. (a) E can be generated overF by some elementy whose minimal polynomial has the specific formTpn−T−z. (b) A formula for the genus ofE is given. (c) IfK is finite, then the genus ofE grows much faster than the number of rational points (as [E∶F] → ∞). (d) We present a new example of a function fieldE/K whose gap numbers are nonclassical.
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